Related papers: Heat kernel asymptotics for magnetic Schr\"odinger…
We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the…
The square root of the heat operator $\sqrt{\partial_t-\Delta}$, can be realized as the Dirichlet to Neumann map of the heat extension of data on $\mathbb R^{n+1}$ to $\mathbb R^{n+2}_+$. In this note we obtain similar characterizations for…
In this article we obtain eigenvalue asymptotics for 2D and 3D-Schroedinger, Schroedinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important. These operators are essentially different and there…
We give an asymptotic expansion of the relative entropy between the heat kernel $q_Z(t,z,w)$ of a compact Riemannian manifold $Z$ and the normalized Riemannian volume for small values of $t$ and for a fixed element $z\in Z$. We prove that…
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties…
In this paper, we study Ornstein-Uhlenbeck operators with quadratic potentials. We use Hamiltonian formalism to characterise the singularities produced by the potentials by finding explicit geodesics of the operators, and obtain the heat…
A new algebraic approach for calculating the heat kernel for the Laplace operator on any Riemannian manifold with covariantly constant curvature is proposed. It is shown that the heat kernel operator can be obtained by an averaging over the…
We calculate the coefficient $a_5$ of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold.
For a given bounded domain $\Omega$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we establish a procedure to get all the coefficients of the asymptotic expansion of the trace of the heat kernel associated with the…
We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schr\"odinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value $b_0$ of the intensity of the magnetic field is strictly…
Let M be a complete Riemannian manifold with a free cocompact Z^k-action. Let k(t,x,y) be the heat kernel on M. We compute the asymptotics of k(t,x,y) in the limit in which t goes to infinity and d(x,y) is comparable to sqrt{t}. We show…
In this paper, we first give a direct proof for two recurrence relations of the heat kernels for hyperbolic spaces in \cite{DM}. Then, by similar computation, we give two similar recurrence relations of the heat kernels for spheres.…
Sharp spectral asymptotics for 2- and 3-dimensional Schroedinger operators with strong magnetic field are derived under rather weak smoothness conditions. In comparison with version 1 of 2005 new results are added and minor errors…
This paper extends results of M. van den Berg on two-term asymptotics for the trace of Sch\"odinger operators when the Laplacian is replaced by non-local (integral) operators corresponding to rotationally symmetric stable processes and…
In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ \Delta^{\alpha/2}_v + v \cdot \nabla_x, \quad \alpha \in (0, 2),\ (x,v)\in {\mathbb…
We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping…
We apply Davies' method for obtaining pointwise lower bounds on the heat kernels of higher-order differential operators to obtain pointwise lower bounds in the presence of a polynomialy bounded potential.
We study magnetic Schr\"odinger operators on graphs. We extend the notion of sparseness of graphs by including a magnetic quantity called the frustration index. This notion of magnetic sparse turn out to be equivalent to the fact that the…
We prove heat kernel bounds for the operator (1 + |x|^{\alpha})\Delta in R^N, through Nash inequalities and weighted Hardy inequalities.
We consider 2-dimensional Schr\"odinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(\mu^{-1}h^{-1})$ or better. We also consider 3-dimensional Schr\"odinger operator…